Number to Power of Zero Rising is One

Theorem

Let $x \in \R$ be a real number.

$x^{\overline 0} = 1$

where $x^{\overline 0}$ denotes the rising factorial.

Proof

\(\ds x^{\overline 0}\)

\(=\)

\(\ds \prod_{j \mathop = 0}^{-1} \paren {x + j}\)

Definition of Rising Factorial

\(\ds \)

\(=\)

\(\ds 1\)

Product is Vacuous

$\blacksquare$